Question: Let $S$ be a piecewise-smooth, closed surface whose interior is denoted $V$. Let $F(x, y, z)$ be a differentiable vector field. Does the divergence theorem necessarily apply to the region $V$, the boundary surface $S$, and the vector field $F$ ? Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
Answer: Assume we have a simple solid region $V$ oriented with outward normals, and it has a piecewise-smooth, closed boundary surface $S$. If $F$ is a continuously differentiable vector field in $\mathbb{R}^3$, then the divergence theorem says: $ \oiint_S F \cdot dS = \iiint_V \text{div}(F) \, dV$ We are never told that the vector field $F$ has continuous derivatives for each of its components, which is what it means to be continuously differentiable. This is a necessary condition of the divergence theorem, so the theorem does not necessarily apply in this case.